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Chapter 9

Chapter 9. Ordinary Differential Equations: Initial-Value Problems Lecture (I) 1. 1 Besides the main textbook, also see Ref.: “Applied Numerical Methods with MATLAB for Engineers and Scientists”, Steven Chapra, 2nd ed., Ch. 20 , McGraw Hill, 2008. Outline. Introduction: Some definitions

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Chapter 9

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  1. Chapter 9 Ordinary Differential Equations: Initial-Value Problems Lecture (I)1 1 Besides the main textbook, also see Ref.: “Applied Numerical Methods with MATLAB for Engineers and Scientists”, Steven Chapra, 2nd ed., Ch. 20, McGraw Hill, 2008. Dr. Jie Zou PHY 3320

  2. Outline • Introduction: Some definitions • Engineering and Scientific Applications • One-step Runge-Kutta (RK) Methods • (1) Euler’s Method • The method (algorithm) • Error analysis (next lecture) • Stability (next lecture) Dr. Jie Zou PHY 3320

  3. Introduction: Some definitions • Differential equation: An equation involving the derivatives or differentials of the dependent variable. • Ordinary differential equation: A differential equation involving only one independent variable. • Example: For the bungee jumper, • Partial differential equation: A differential equation involving two or more independent variables (with partial derivatives). • Order of a differential equation: The order of the highest derivative in the equation. • Example: For an unforced mass-spring system with damping-a second-order equation: Dr. Jie Zou PHY 3320

  4. Introduction: Some definitions (cont.) • For an nth-order differential equation, n conditions are required to obtain a unique solution. • Initial-value problem: All conditions are specified at the same value of the independent variable (e.g., at x or t = 0). • Example: For the bungee jumper, • Boundary-value problem: Conditions are specified at different values of the independent variable. • Example: Particle in an infinite square well Initial Condition Fig. PT6.3 (Ref. by Chapra): Solutions for dy/dx = -2x3 + 12x2 – 20x + 8.5 with different constants of integration, C. Boundary Conditions

  5. Engineering and scientific applications Fig. PT6.1 (Ref. by Chapra): The sequence of events in the development and solution of ODEs for engineering and science. Dr. Jie Zou PHY 3320

  6. Euler’s method • Let’s look at the Bungee-Jumper’s example: Solve an ODE-initial-value problem (1) • Step 1: Finite-difference approximation for dv/dt (2) • Step 2: Substitute Eq. (2) in Eq. (1) • (3) • Step 3: Notice that dv/dt at ti = g-cdv(ti)2/m, (3) becomes Fig. 1.4 (Ref. by Chapra): Numerical solution by Euler’s method. Euler’s method (a one-step method)

  7. Another look at Euler’s method • Solving ODE: dy/dt = f(t,y) • All one-step methods (Runge-Kutta methods) have the general form: • : an increment function for extrapolating from an old value yi to a new value yi+1. • One-step methods: use information from one pervious point i to extrapolate to a new value. • h: Step size = ti+1 – ti. • Euler’s method: •  = f(ti,yi), the 1st derivate of y at ti • yi+1 = yi + f(ti,yi)h Fig. 20.1 (Ref. by Chapra): Euler’s method Dr. Jie Zou PHY 3320

  8. Example: Euler’s method • Example 20.1 (Ref.): Use Euler’s method to integrate y’ = 4e0.8t – 0.5y from t = 0 to t = 4 with a step size of 1. The initial condition at t = 0 is y = 2. Note that the exact solution can be determined analytically as y = (4/1.3)(e0.8t – e-0.5t) + 2e-0.5t Dr. Jie Zou PHY 3320

  9. Results Table 20.1 (Ref. by Chapra) Fig. 20.2 (Ref. by Chapra) Dr. Jie Zou PHY 3320

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